President Trump and North Korean leader Kim Jong Un signed a document that Mr. Trump described as "important" and "comprehensive," to conclude their meetings in Singapore. The document, which offers few details about how the aims will be accomplished, says North Korea commits to working towards denuclearization.

Mr. Trump said he "absolutely" will invite Kim to the White House. Mr. Trump said he thinks the summit went "better than anybody could have expected."

As they began the summit, Mr. Trump said he thinks he and Kim will have a "terrific relationship," and Kim, through a translator, said North Korea had to overcome a number of "obstacles" to get to this moment.

Mr. Trump and Kim participated in a one-on-one meeting, with translators only, followed by an expanded meeting including their top advisers, and a working lunch.

Read live updates below as they happened at the Kim Jong Un-Trump summit:

Trump suggests U.S. will cease military exercises with South Korea for now

Mr. Trump said it would be "inappropriate" for the U.S. to perform joint military exercises with South Korea while working out this deal with North Korea, declaring such "war games" to be "provocative."

Sanctions remain until nukes are gone, Trump says

Mr. Trump said sanctions will remain in place until the nukes are gone. The sanctions will go away "once we can be sure the nukes are no longer a factor," he said.

Trump claims Kim has accepted invitation to White House

Mr. Trump not only claimed he invited Kim to the White House, but claimed Kim has "accepted" that invitation.

Trump says human rights were "discussed" but only "briefly"

Mr. Trump said human rights were brought up during his conversation with Kim.

Pompeo, high-level North Korean official to continue conversation

According to the document Mr. Trump and Kim signed, Secretary of State Mike Pompeo and a high-level North Korean official will hold "follow-up" negotiations at the earliest possible date, "to implement the outcomes of the U.S.-DPRK summit."

The document Mr. Trump and Kim signed lists four provisions, including reaffirming North Korea's "commitment to working toward complete denuclearization of the Korean peninsula."

The White House has yet to officially release the document, but show-up photos of the document reveal some of its contents.

Here are the four components of agreement outlined in the document:

"The United States and the DPRK commit to establish new U.S.-DPRK relations in accordance with the desire of the peoples of the two countries for peace and prosperity."

"The United States and the DPRK will join their efforts to build a lasting and stable peace regime on the Korean peninsula."

"Reaffirming the April 27, 2018 Panmunjom Declaration, the DPRK commits to work toward complete denuclearization of the Korean peninsula."

"The United States and the DPRK commit to recovering POW/MIA remains, including the immediate repatriation of those already identified."

A joint document signed by President Donald Trump and North Korean leader Kim Jong Un is seen in the hands of Mr. Trump immediately after the two men signed it following a summit on Sentosa Island, Singapore, June 12, 2018.

Getty

Trump says Kim has a great personality, says they will meet many times

Mr. Trump, speaking to reporters briefly, was asked what surprised him most about Kim.

"Great personality and very smart -- good combination," the president responded.

Mr. Trump also declared Kim a very worthy negotiator.

"I learned that he's a very talented man. I also learned that he loves his country very much," he said.

This summit isn't the last, said Mr. Trump who said he expects to meet with Kim "many" times.

Trump, Kim sign "important" and "comprehensive" document

Mr. Trump, seated at a table alongside Kim, said they are signing an "important" and "pretty comprehensive document."

"Today we had a historic meeting and decided to leave the past behind," Kim said through a translator, at the signing table. "The world will see a major change."

Asked if North Korea agreed to denuclearize, Mr. Trump said they would be starting that process quickly.

Room set up for signing event

It's still unclear what exactly Mr. Trump and Kim will be signing, but the room for the signing has been set up.

Kim appears to check out Trump's vehicle

After Mr. Trump said they would be signing something, the president appeared to show Kim his vehicle.

It is unclear what is happening next.

Trump says he and Kim will be "signing" something

When Mr. Trump and Kim departed from their lunch, Mr. Trump said they will be "signing" something.

It is unclear what that might be, although Mr. Trump said the public will find out soon.

Trump-Kim summit hits two-hour mark

Now that it's after 11 p.m. EDT, or 11 a.m. in Singapore, the Trump-Kim summit has reached the two-hour mark.

The expanded bilateral meeting is ongoing, and is still expected to last a while.

Who is Trump's translator?

Mr. Trump's translator, according to a U.S. State Department official, is Yun-hyang Lee, ‎Division Chief of Interpreting Services at the State Department.

Translators in a situation like this one, whether neither Mr. Trump nor Kim is fluent in the other's language and where the stakes are this high, are key.

Kim's translator: Many will see meeting as a "science fiction movie"

The translator for Kim to Mr. Trump, ahead of the expanded bilateral meeting, said many people will think of the summit "as a science fiction movie."

"Many people in the world will think of this as a (inaudible) form of fantasy...from a science fiction movie," the translator said.

Trump says that, "working together, we will get it taken care of"

Here is what was said at the top of an expanded bilateral meeting between Mr. Trump and Kim, and their top officials.

TRUMP: "Working together, we will get it taken care of."

KIM: "There will be challenges ahead but we will work with Trump. We overcame all kinds of skepticism and speculations about this summit and I believe that this is good for the peace."

TRUMP: "We will solve it...and I look forward to working on it with you. Thank you very much. Thank you very much everybody. Thank you."

President Trump (3rd R) shakes hands with North Korean leader Kim Jong Un (3rd L) as they sit down with their respective delegations for the U.S.-North Korea summit, at the Capella Hotel on Sentosa island in Singapore on June 12, 2018.

Getty

Trump says meeting is "very, very good" so far

After their one-on-one meeting, Mr. Trump and Kim sat down for an expanded bilateral meeting.

In the room representing the U.S. are chief of staff John Kelly, Secretary of State Mike Pompeo and national security adviser John Bolton.

Ahead of that expanded bilateral meeting, Mr. Trump was asked how their summit is going so far.

"Very good," Mr. Trump said. "Very, very good. We have a great relationship. Thank you, thank you very much."

Trump and Kim complete first conversation

After about 35 minutes, at roughly 9:50 p.m., Mr. Trump and Kim emerged from the one-0n-one meeting.

They are now expected to proceed to an expanded bilateral meeting.

Trump and Kim speak, kicking off what Trump says will be a "terrific" relationship

Mr. Trump sat next to Kim and spoke ahead of their meeting.

"I feel really great," Mr. Trump told reporters, adding that it's his "honor" to meet and he and Kim will have a "terrific relationship."

"I feel really great, we're going to have a great discussion, and I think tremendous success," Mr. Trump said. "It will be tremendously successful. And its my honor, and we will have a terrific relationship I have no doubt."

Kim, through his translator, said there were a lot of obstacles to get to this moment, but they made it.

Only translators are expected to be in the meeting with Mr. Trump and Kim.

Trump, Kim shake hands for the first time

Mr. Trump and Kim shook hands for the first time just moments after 9 a.m. Mr. Trump did not smile as they shook hands, although Kim was spotted smiling after they greeted.

They shook hands for several seconds, before walking off towards their meeting together.

Trump, Kim have both arrived at summit site

By 9 p.m. EDT, both Mr. Trump and Kim had arrived at the site of the summit.

Kim Jong Un arrives at the Capella Hotel in Singapore for his meeting with President Trump on Tuesday, June 12, 2018.

CBS News

Trump tweets that top adviser Larry Kudlow suffered a heart attack

With moments to go before the Trump-Kim summit, the president tweeted that his top economic adviser, Larry Kudlow, has suffered a heart attack and is at Walter Reed National Military Medical Center in Washington, D.C.

Our Great Larry Kudlow, who has been working so hard on trade and the economy, has just suffered a heart attack. He is now in Walter Reed Medical Center.

A live look at Trump and Kim's hotels as they prepare to depart for the meeting

Cameras are stationed outside the hotels where Mr. Trump and Kim are staying. They're expected to depart their respective locations at 8 p.m. ET for the drive to the Capella Hotel, where the meeting will take place.

Trump speaks with South Korean and Japanese leaders ahead of summit

The White House says the president spoke on the phone with South Korean President Moon Jae-in and Japanese Prime Minister Shinzo Abe on Monday ahead of the summit.

Here's how the White House describes the call with Moon:

President Donald J. Trump spoke yesterday with President Moon Jae-in of the Republic of Korea to discuss recent developments ahead of today's summit with North Korea. The two leaders vowed to continue their close coordination following President Trump's meeting with North Korean leader Kim Jong Un in Singapore.

And with Abe:

President Donald J. Trump spoke yesterday with Prime Minister Shinzo Abe of Japan. The two leaders discussed recent developments ahead of today's summit with North Korea and agreed to consult closely following the meeting.

Trump tweets at "haters & losers" before meeting

A few hours before his meeting with Kim, the president acknowledged in a tweet that preliminary meetings "between staffs and representatives are going well and quickly," adding, "but in the end, that doesn't matter. We will all know soon whether or not a real deal, unlike those of the past, can happen!"

Meetings between staffs and representatives are going well and quickly....but in the end, that doesn't matter. We will all know soon whether or not a real deal, unlike those of the past, can happen!

Then, he slapped at his critics, "the haters & losers" who say that "the fact that I am having a meeting [with Kim] is a major loss." He argued, "We have our hostages, testing, research and all missle [SIC] launches have stoped [SIC], and these pundits, who have called me wrong from the beginning, have nothing else they can say! We will be fine!"

What is success for the North Korea summit?

Pompeo told reporters Monday that if diplomacy does not move in the right direction after Tuesday's talks, pressure will be ratcheted up on the North Korean regime. "We will see how far we get," Pompeo said of the negotiations.

Pompeo also emphasized that the U.S. has its eyes open, since it has been "fooled" before during prior attempts to reach peace in the region. He says that the "v" - verification -matters most.

What's up for discussion?

Both Mr. Trump and Kim are expected to largely focus their discussion on Tuesday around the denuclearization of the Korean Peninsula. Pompeo told reporters on Monday that the U.S. is prepared to offer "different, unique" security assurances to North Korea.

"We're prepared to take what will be security assurances that are different, unique than have been provided - than America has been willing to provide previously," Pompeo said. The secretary added that he's hopeful the summit will set the conditions for future productive talks and a potential framework for work towards denuclearization.

The president told Prime Minister Lee Hsien Loong of Singapore on Monday that he thinks "things could work out very nicely" during Tuesday's talks.